Statistics of the two-star ERGM

Mukherjee, Sumit; Xu, Yuanzhe

doi:10.48550/arxiv.1310.4526

Cited by 2 publications

(1 citation statement)

References 25 publications

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“…This particular ERGM is chosen for two reasons. Firstly, it covers the currently most complex ERGM for which a thorough theoretical analysis for parameter estimation is available, see Mukherjee and Xu [2013]. Secondly, there are not many competing approaches available for our task and some of them are devised specifically for ERGMs.…”

Section: Synthetic Experimentsmentioning

confidence: 99%

AgraSSt: Approximate Graph Stein Statistics for Interpretable Assessment of Implicit Graph Generators

Xu¹,

Reinert²

2022

Preprint

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We propose and analyse a novel statistical procedure, coined AgraSSt, to assess the quality of graph generators that may not be available in explicit form. In particular, AgraSSt can be used to determine whether a learnt graph generating process is capable of generating graphs that resemble a given input graph. Inspired by Stein operators for random graphs, the key idea of AgraSSt is the construction of a kernel discrepancy based on an operator obtained from the graph generator. AgraSSt can provide interpretable criticisms for a graph generator training procedure and help identify reliable sample batches for downstream tasks. Using Stein's method we give theoretical guarantees for a broad class of random graph models. We provide empirical results on both synthetic input graphs with known graph generation procedures, and real-world input graphs that the state-of-the-art (deep) generative models for graphs are trained on.

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Section: Synthetic Experimentsmentioning

confidence: 99%

AgraSSt: Approximate Graph Stein Statistics for Interpretable Assessment of Implicit Graph Generators

Xu¹,

Reinert²

2022

Preprint

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Sub-critical exponential random graphs: concentration of measure and some applications

Ganguly¹,

Nam²

2024

Trans. Amer. Math. Soc.

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The exponential random graph model (ERGM) is a central object in the study of clustering properties in social networks as well as canonical ensembles in statistical physics. Despite some breakthrough works in the mathematical understanding of ERGM, most notably by Bhamidi, Bresler, and Sly [Ann. Appl. Probab. 21 (2011), pp. 2146–2170] through the analysis of a natural Heat-bath Glauber dynamics, and by Chatterjee and Diaconis [Ann. Statist. 41 (2013), pp. 2428–2461; Eldan [Geom. Funct. Anal. 28 (2018), pp. 1548–1596; and Eldan and Gross [Ann. Appl. Probab. 28 (2018), pp. 3698–3735] via a large deviation theoretic perspective, several basic questions have remained unanswered owing to the lack of exact solvability unlike the much studied Curie-Weiss model (Ising model on the complete graph). In this paper, we establish a series of new concentration of measure results for the ERGM throughout the entire sub-critical phase, including a Poincaré inequality, Gaussian concentration for Lipschitz functions, and a central limit theorem. In addition, a new proof of a quantitative bound on the W 1 − W_1- Wasserstein distance to Erdős-Rényi graphs, previously obtained by Reinert and Ross [Ann. Appl. Probab. 29 (2019), pp. 3201–3229], is also presented. The arguments rely on translating temporal mixing properties of Glauber dynamics to static spatial mixing properties of the equilibrium measure and have the potential of being useful in proving similar functional inequalities for other Gibbsian systems beyond the perturbative regime.

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Statistics of the two-star ERGM

Cited by 2 publications

References 25 publications

AgraSSt: Approximate Graph Stein Statistics for Interpretable Assessment of Implicit Graph Generators

AgraSSt: Approximate Graph Stein Statistics for Interpretable Assessment of Implicit Graph Generators

Sub-critical exponential random graphs: concentration of measure and some applications

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